I normally don't like to compare brands but it seems that this topic will remain hotly contested regardless of what any single blog or person says.
Vacheron Constantin (1755) is the oldest of the three, followed by Patek (1839), and then AP (1875) but for the purpose of this comparison, we'll largely disregard history and try to evaluate the three based on merit.
The modern Vacheron Constantin, Audemars Piguet, and Patek Philippe (to a lesser extent) uses a large number of Jaeger-LeCoultre movements (AP 2124=VC 1126=JLC 889, VC 1120=JLC 920, AP 2003=VC 1003 which is similar to JLC 849) as well as chronograph movements from Lemania and Frederique Piguet (not related to AP). There's nothing wrong with using movements from either JLC, FP, or Lemania but in the interest of being more exclusive, in-house movements are preferred. This being said, a well finished Lemania movement is a work of art that trumps most in house movements.
VC 1141, based on Lemania 2320 also previously used by Patek
Patek 29-535, PP's successor to the Lemania 2320 (Cal. 27-70)
Patek uses a much higher proportion of in house movements, with many innovative features such as silicon balance wheel and hairspring. One of my personal favorite is the Spiromax balance wheel weights which not only offer the precision of a free sprung balance but is also quite elegant.
Patek silicon escapement and hairspring
with Spiromax balance wheel
Patek definitely is the winner in terms of movements. Its movements are superiorly finished, the technology in them is superb, and the fact that they're increasingly in-house is just the cherry on top. They also have probably the most consistent lineup of watches that remain uniquely identifiable despite their traditional appearance.
Patek Cal. 31-260 used in the Regulator Annual Calendar
Patek, however, has not been the most artistic brand. Its traditional case designs makes comparison with A. Lange & Sohne inevitable and I see Patek as the more artistic, less mechanical, and more soulful competitor to Lange. Patek cases are superbly made but there is a lack of the art found in VC and AP.
The Datograph is an inevitable contender to Patek
VC has a history of modifying and finishing movements, but I see the greatest merit of VC as its ability to tastefully execute artistic watch designs. Even beyond the Métiers d'Art collection, VC is not afraid to make larger watches with exquisite detail.
Vacheron Constantin Métiers d'Art Chinese Zodiac: Snake
Having been to the manufacturer in Geneva, I can say that VC is no slouch when it comes to movements either, especially in their atelier workshop where the high complications are made. Its ability to create commissioned pieces is also quite unique, while this has been historically possible with Patek, it's much more difficult than with VC.
Movement in the VC Quai D'Ile with its unique bridge
Beautiful but perhaps a bit hard to read
The flip side of the art, however, is that the average VC is less well finished than the average Patek. Although VC's art usually doesn't intervene on timekeeping, I'm quite unhappy whenever one could misread the hour. I'm also not a fan of overly large watches and prefer that everything (including grande complications) stay under 40 mm (maybe 42 mm for a minute repeater, perpetual calendar, chronographe, equation of time), something that VC has horribly violated with its 47mm Vladimir custom watch which would have otherwise been an immaculate work of art.
Immaculate bas relief...
...executed on a 47mm monstrosity
AP is a brand that I want to love and really appreciate on an intellectual level but have been unable to. It has some extremely unique pieces like the Millenary that VC and PP would not dream of making. The Royal Oak has become probably the most iconic of the trinity of the VC Overseas, PP Nautilus, and AP Royal Oak. The in house 3120 movement also deserves all the accolade it's garnishing.
But the reason I have been unable to love AP is its size and strange approach to movements. The Royal Oak Offshore has been described as a "Ferarri with a four cylinder engine" due to its modular chronograph design. While modular designs function similarly to integrated designs, service is more difficult and the design is far less elegant with the pushers sitting either above or below the crown. The AP lineup is also inconsistent and while the Millenary and Royal Oak lines are easily recognizable, the Edward Piguet and Jules Audemars lines are strange looking with no clear direction. If AP is to steer clear of the path that Blancpain has taken and not become a one watch brand (AP Royal Oak and Blancpain Fifty Fathom), it needs some serious work on its dress watches.
Overall I see Patek as a company seeking to make consistent, conservative, and mechanically innovative watches with a weaknesses for art. Vacheron as a company able to merge art and mechanics with a weaknesses for movements. Audemars Piguet as an innovative company that is improving its movements but suffers from the occasional oddball rubber-clad watch with a chronograph module on someone else's movement. All three are meritorious brands deserving due consideration. Which one wins for me? A. Lange & Sohne of course.